Matrix Inequality Problem Research Articles

New exponential stability criterion for switched linear systems with average dwell time

In this article, the partition of state space is investigated based on the average dwell time theory, and a novel switching condition is proposed. Under the switching condition and strategy, a switched system where all subsystems are unstable is globally exponentially stabilized. Compared with the existing literature, the conclusion is less conservative and more general. Consequently, the conditions appropriate for this exponential stability issue are derived and a computational procedure to solve the bilinear matrix inequality problems and admissible average dwell time is provided. Finally, two numerical examples are used to show the validity and effectiveness of the proposed methods.

Distributed adaptive model‐based event‐triggered predictive control for consensus of multiagent systems

SummaryThis paper investigates the consensus problem for discrete‐time networked multiagent systems, where the information is exchanged through a shared network with communication delays. Based on the event‐triggered communication scheme together with the networked predictive control scheme, a novel distributed adaptive model‐based event‐triggered predictive control protocol is proposed to reduce the network communication burden and compensate for the communication delays actively. By using the virtual interval segmentation technique and the time‐delay modeling method together with defining a set of switching signals, the closed‐loop systems are established in forms of switching time‐delay systems. The problem of simultaneously deriving the controller gain matrix and the event‐triggering parameter matrices is cast into standard linear matrix inequalities problem. A convincing simulation example is given to demonstrate the theoretical results.

Robust Fuzzy Filter Design for Nonlinear Parabolic Partial Differential Systems with Continuous Wiener Noise and Discontinuous Poisson Noise

This paper addresses a robust \(H_{\infty }\) fuzzy filter design problem for nonlinear stochastic partial differential systems (NSPDSs) with continuous random fluctuation, discontinuous Poisson jumping noise, random external disturbance and measurement noise in the spatiotemporal domain. For NSPDSs, the robust \(H_{\infty }\) filter design problem through a measurement output needs to solve a complex second-order Hamilton Jacobi integral inequality. In order to simplify the design procedure, a fuzzy stochastic partial differential system based on a fuzzy interpolation approach is proposed to approximate the NSPDS. Then, the robust \(H_{\infty }\) fuzzy filter design problem can be reformulated as a diffusion matrix inequality (DMI) problem. Since the DMI problem is difficult to be solved via traditional algebraic techniques, we utilize the divergence theorem and Poincare inequality to transform the DMIs to a set of linear matrix inequalities (LMIs) which could be easily solved with the help of MATLAB LMI Toolbox. Finally, a robust state estimation problem of an ecology system with intrinsic spatiotemporal continuous Wiener noise and discontinuous Poisson jump fluctuation is provided as an example to illustrate the design procedure and to confirm the \(H_{\infty }\) filtering performance of the proposed \(H_{\infty }\) fuzzy filter design method.

Robust H∞ gain neuro-adaptive observer design for nonlinear uncertain systems

To guarantee convergent state estimates and exact approximations, it is highly desirable that observers can independently dominate the effects of unmodelled dynamics. Based on adaptive nonlinear approximation, this paper presents a robust H∞ gain neuro-adaptive observer (R H∞GNAO) design methodology for a large class of uncertain nonlinear systems in the presence of time-varying unknown parameters with bounded external disturbances on the state vector and on the output of the original system. The proposed R H∞GNAO incorporates radial basis function neural networks (RBFNNs) to approximate the unknown nonlinearities in the uncertain system. The weight dynamics of every RBFNN are adjusted online by using an adaptive projection algorithm. The asymptotic convergence of the state and parameter estimation errors is achieved by using Lyapunov cogitation under a well-defined persistent excitation condition, and without recourse to the strictly positive real condition. The repercussions of unknown disturbances are reduced by integrating an H∞ gain performance criterion into the proposed estimation approach. The condition imposed by this proposed observer approach, such that all estimated signals are uniformly ultimately bounded, is expressed in the form of the linear matrix inequality problem and warrants the demanded performances. To evaluate the performance of the proposed observer, various simulations are presented.

Robust integral-observer-based fault estimation for Lipschitz nonlinear systems with time-varying uncertainties

This paper addresses the problem of state estimation and sensor fault reconstruction conjointly for a class of nonlinear systems with time-varying uncertainties for which the nonlinear characteristic satisfies the Lipschitz circumstance. A hybrid approach based on an integral observer and sliding-mode theory has been proposed in order to model sensor fault as a virtual actuator one. For the augmented model, the observer matching condition is not satisfied. To overcome this problem, a new method, which improves the design approach and enhances the rapidity of the fault estimation convergence, has been proposed. The fault estimation error effect is minimized by integrating the [Formula: see text] disturbance attenuation level. The proposed design is formulated and derived as a linear matrix inequality problem. Parameters of this observer are calculated through the linear matrix inequality technique. The proposed method has been validated through an example of a single-link manipulator robot. Simulation results show that this approach can estimate the state and the sensor fault successfully, despite the time-varying uncertainties and the presence of unknown inputs.

Adaptive composite non‐linear feedback‐based sliding mode control for non‐linear systems

An adaptive composite non-linear feedback based sliding mode controller is proposed for a class of non-linear systems comprising uncertainty terms. In these cases, it is theoretically proved that the trajectory of the closed-loop system converges to the sliding surface in a finite-time manner. Also, by the means of numerical linear matrix inequality technique, the stability of the non-linear system along the sliding surface is turned to an linear matrix inequality problem. By solving such problem, the unknown matrix of the sliding surface and composite non-linear feedback based sliding mode controller is obtained. Finally, to show the merits of the proposed approach, it is applied to practical non-linear DC servomotor.

Second-order sliding set design for a class of uncertain nonlinear systems with disturbances: An LMI approach

In this paper, a linear matrix inequality-based second-order sliding set control is proposed for the uncertain systems with time-varying uncertainties, nonlinearities and external disturbances. The linear matrix inequality conditions based on the Lyapunov stability theory are calculated which guarantee the robustness against external disturbances and changes of internal parameters. These conditions certify the asymptotic convergence of the state trajectories to origin during the sliding phase, and designed switching controller ensures the states reach the sliding surface. It is presented that the stability analysis can simplify the nonlinear system into a linear matrix inequality problem. The simulation results are displayed to show the effectiveness of the proposed approach.

Experimental interval models for the robust Fault Detection of Aircraft Air Data Sensors

In this paper data-based approaches for a robust Fault Detection (FD) of the Air Data Sensors (ADS) including airspeed angles of attack and sideslip are proposed. Experimental Interval Models (IMs) have been considered for coping with modeling uncertainty and for providing interval estimations of the ADS signals. Specifically, a nonlinear-in-the-parameter Neural Network model has been introduced to characterize the nominal nonlinear response in the different phase of the flight, while model uncertainty is captured by an additional additive contribution provided by a linear in the parameters IM. The FD is achieved by verifying whether the measured ADS signals fall within time-varying bounds predicted by the nonlinear + IM. The IM identification has been formalized as a Linear Matrix Inequality (LMI) problem using as cost function the mean amplitude of the prediction interval and, as optimization variables, the amplitudes of the uncertain parameters of the model. The model identification was based on multi flight experimental data of a P92 Tecnam aircraft. The proposed method is compared with conventional FD schemes with fixed thresholds. Extensive validation tests have been conducted by injecting artificially additive hard and incipient failures on the ADS. The FD scheme has shown to be remarkably robust in all phases of the flight in terms of low false alarm rates while maintaining desirable detectability to faults.

Sampled-data static output-feedback control for nonlinear systems in T–S form via descriptor redundancy

Abstract This paper deals with a static output-feedback stabilization problem within a sampled-data framework for nonlinear systems in Takagi–Sugeno form. We investigate this problem on the basis of the descriptor-redundancy scheme. The resulting features are that: (i) the sampled-data synthesis does not involve (even approximate) discrete-time models of nonlinear systems; (ii) the sampled-data stability is analyzed in the continuous-time Lyapunov sense; and (iii) the design algorithm consists of a single-stage linear matrix inequality problem without linear matrix equality constraints.

Global asymptotic stability assessment of three‐phase inverters with saturation

Saturation is reported to have a significant influence on the stability of the inverter. In this study, a direct Lyapunov function method to analyse the stability of a three-phase inverter with saturation is presented. The overall model of the three-phase inverter is established and the order of the original model is reduced based on the singular perturbation theory. Then the global asymptotic stability assessments of inverters are transformed into linear matrix inequalities problems that are easy to be solved numerically. Through the analysis of this study, it can be proved that the three-phase inverter based on conventional proportional-integral modulation can achieve global stabilisation. The proportional coefficients of the voltage loop have the greatest influence on the stability of the three-phase inverter. Besides, the saturation also has an important effect on the stability of the inverter. The proposed method has some advantages compared with the small-signal analysis. It can quantify the stability degree of the inverter and explain some non-linear phenomena caused by saturation. Simulation developed in PSCAD demonstrates the effectiveness of this method and the results are compared with the real-time digital simulation results implemented in RT-LAB and hardware results.

Sequential parametric convex approximation algorithm for bilinear matrix inequality problem

The goal of this paper is to study algorithms for solving optimization problems subject to bilinear matrix inequalities (BMIs). This class of problems is known to be of great importance in engineering applications, for instance, control system designs. A main contribution is the development of a sequential convex optimization algorithm, where at each iteration step, a convex subproblem with linear matrix inequality (LMI) constraints is solved. The set of feasible points of the LMIs is a convex inner approximation of the set of feasible points of the BMI constraints around the current iteration point. Another contribution is the convergence proof of a subsequence of the iterations to a stationary point. Finally, an example of the static output-feedback controller design problem is given for comparative analysis.

Line-Independent Plug-and-Play Controllers for Voltage Stabilization in DC Microgrids

We consider the problem of stabilizing voltages in Direct Current (DC) microgrids given by the interconnection of Distributed Generation Units (DGUs), power lines, and loads. We propose a decentralized control architecture where the primary controller of each DGU can be designed in a plug-and-play fashion, allowing the seamless addition of new DGUs. Differently from several other approaches to primary control, local design is independent of the parameters of power lines and the only global quantity used in the synthesis algorithm is a scalar parameter. Moreover, differently from the plug-and-play control scheme in [1] , the plug-in of a DGU does not require to update the controllers of neighboring DGUs. Local control design is cast into a linear matrix inequality problem that, if infeasible, allows one to deny plug-in requests that might be dangerous for microgrid stability. The proof of closed-loop stability of voltages exploits structured Lyapunov functions, the LaSalle invariance theorem and the properties of graph Laplacians. Theoretical results are backed up by simulations in PSCAD.

Event‐triggered constrained control of positive systems with input saturation

SummaryThis paper addresses the problem of event‐triggered stabilization for positive systems subject to input saturation, where the state variables are in the nonnegative orthant. An event‐triggered linear state feedback law is constructed. By expressing the saturated linear state feedback law on a convex hull of a group of auxiliary linear feedback laws, we establish conditions under which the closed‐loop system is asymptotically stable with a given set contained in the domain of attraction. On the basis of these conditions, the problem of designing the feedback gain and the event‐triggering strategy for attaining the largest domain of attraction is formulated and solved as an optimization problem with linear matrix inequality constraints. The problem of designing the feedback gain and the event‐triggering strategy for achieving fast transience response with a guaranteed size of the domain of attraction is also formulated and solved as an linear matrix inequality problem. The effectiveness of these results is then illustrated by numerical simulation.

Static Output Feedback Stabilization of a Class of Switched Linear Systems with State Constraints

This paper will research the problem of static output feedback (SOF) stabilization of state-constrained switched linear systems via an improved average dwell time method (ADT). Firstly, an improved ADT method is adopted to establish sufficient conditions for SOF of the state-constrained switched linear systems in the form of matrix inequality. It has been shown that this method is less conservative than traditional ADT, which in view of different decay rates of a Lyapunov function related to an active subsystem on the basis of whether the saturations occur or not. Then, a new iterative algorithm is designed to solve the matrix inequality and a SOF controller can be added. In the iterative linear matrix inequality (ILMI) algorithm, it is important not only to overcome the typical bilinear matrix inequality (BMI) problem of SOF, but also to solve the non-convex problem caused by state constraints. Finally, the availability and the applicability of the proposed method is shown by the application of a boost converter.

Trajectory tracking for two-degree of freedom helicopter system using a controller-disturbance observer integrated design

Trajectory tracking control for helicopters, which are widely used in severe situations such as military and rescue missions, is a challenging field of research. In helicopter system, the stability problem and predefined trajectories tracking are main challenges, especially in the presence of external disturbances and dynamic model uncertainties. Hence, a robust control design is needed for tracking the desired references. There has been a lot of motivation for solving these problems with simpler methods and also reducing the couplings in the helicopter system to achieve better performance, as the presented paper attempts to fill these gaps. This paper focuses on designing control laws for two-degree of freedom helicopter system while assuring the closed-loop stability. A nonlinear disturbance observer-based control (NDOBC) is designed for attenuating the effects of exogenous disturbances. Trajectory tracking controller and nonlinear disturbance observer are formulated in the form of two linear matrix inequality (LMI) problems. The closed-loop system stability, including controller and observer, is investigated by Lyapunov theorem. The effectiveness of the proposed design for tracking the trajectories (vertical flight and pitch angle rotor blade) and disturbance estimation is verified by simulation results.

Sequential Convex Overbounding Approximation Method for Bilinear Matrix Inequality Problems

Abstract This paper concerns with the overbounding method for solving the bilinear matrix inequality (BMI) problem. The overbounding method is essentially the same as the difference of convex optimization approach and the inner approximation method. The unified view of these methods is provided. This paper also proposes a new overbounding algorithm based on the less conservative overbounding approximation and the matrix decomposition.

Design of a feedback control law for steering steady-states of biochemical reaction networks in quasi-polynomial framework

This paper presents a design of stabilizing dynamic feedback control law for steering steady-states of biochemical reaction networks. The design is carried out on a structurally similar reduced linear system model, obtained at the neighborhood of equilibrium point of the nonlinear system. The biochemical reaction network is modeled as a quasi-polynomial system with a translation into the positive orthant retaining all qualitative behaviors of the original system. The control law design is formulated as a bilinear matrix inequality (BMI) problem and an iterative algorithm is presented to find the controller gains. Two examples are illustrated to show the effectiveness of the proposed method.

Coordinated design and application of robust damping controllers for shunt FACTS devices to enhance small-signal stability of large-scale power systems

This paper presents a robust and coordinated supplementary damping controller design of multiple reactive FACTS controllers and their application in a large-scale power system. Reactive FACTS devices, such as static synchronous compensators (STATCOM) and static VAR compensators (SVC), are considered and assessed for their damping controller design. Control objectives, including regional pole placement and norm bounded mix sensitivities, are used to solve the bilinear matrix inequality (BMI) problems in each linearized model via a two-step method. Multiple damping controllers are sequentially designed to avoid coupling effect among the input signals and increase the reliability of the proposed design. A 5-area 16-machine 68-bus power system is used for the implementation of the damping controllers. Numerical linear analysis and real-time simulations in a test platform based on Real-time digital simulators (RTDS) are adopted to test the feasibility and robustness of the coordinated damping controllers.

Robust Fuzzy-Based Sliding Mode Control for Uncertain Discrete Nonlinear Systems for Achieving Performance Requirements

A novel sliding mode fuzzy controller design method is investigated in this paper for uncertain stochastic nonlinear systems described by the Takagi–Sugeno fuzzy model. Applying integral sliding control scheme, a reaching controller is developed such that the trajectory of the system is approximated to the assigned sliding surface to guarantee robustness. Next, a fuzzy controller is designed by the concept of parallel distributed compensation to achieve multiple performance constraints, including individual state variance constraint and strictly input passivity. Besides, some sufficient conditions are derived into linear matrix inequality problem via combining Lyapunov stability criterion, passivity theory, and covariance theory. Using convex optimization algorithm, the required feasible solutions can be directly obtained by solving the derived conditions. Thus, a sliding mode fuzzy controller can be established such that uncertain nonlinear stochastic system achieves robustness, stability, individual state variance constraint, and strictly input passivity at the same time. At last, a simulation result of controlling nonlinear ship steering system is proposed to demonstrate the usefulness and applicability of this paper.

Admissibilization of Singular Interval-Valued Fuzzy Systems

In this paper, we consider the state-feedback and static output feedback control problems of continuous singular interval-valued Takagi–Sugeno fuzzy systems with mismatched membership functions. First, based on the Lyapunov stability theory, a state-feedback control criterion is proposed to guarantee the closed-loop system to be admissible. Second, the result is extended to the static output feedback control problem, and a sufficient condition for synthesis is derived in terms of strict linear matrix inequalities, which eliminates the disadvantages in some existing results, such as same output matrices and the bilinear matrix inequality problem. To obtain less conservative results, the information of mismatched membership functions is employed. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.